For what value of $k$ does the line represented by the equation $1-kx = -3y$ contain the point $(4,-3)$?
Since $(4, -3)$ lies on the line, we plug $x = 4$ and $y = -3$ into the equation to get $1 - 4k = -3\cdot -3 \Rightarrow k = \boxed{-2}$.